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This paper presents a novel method for volume rendering of unstructured grids. Previously, we introduced an algorithm for perspective-correct interpolation of barycentric coordinates and computing polynomial attenuation integrals for a projected tetrahedron using graphics hardware. In this paper, we enhance the algorithm by providing a simple and efficient method to compute the projected shape (silhouette) and tessellation of a tetrahedron, in perspective and orthographic projection models. Our tessellation algorithm is published for the first time. Compared with works of other groups on rendering unstructured grids, the main contributions of this work are: 1) A new algorithm for finding the silhouette of a projected tetrahedron. 2) A method for interpolating barycentric coordinates and thickness on the faces of the tetrahedron. 3) Visualizing higher-order attenuation functions using GPU without preintegration. 4) Capability of applying shape deformations to a rendered tetrahedral mesh without significant performance loss. Our visualization model is independent of depth-sorting of the cells. We present imaging and timing results of our implementation, and an application in time-critical "2D-3D" deformable registration of anatomical models. We discuss the impact of using higher-order functions on quality and performance.